More than sixty years ago, Mackey [1]} asked whether
density operators represent the most general notion of a quantum state
that is consistent with the standard description of observables as
self-adjoint operators. Gleason [2]} responded with
a proof that—in separable Hilbert spaces of dimension greater than
two—every state must admit an expression in terms of a density
operator if it is to consistently assign probabilities to the measurement
outcomes of such observables. In 2003, Busch [3]} (and
then Caves et al. [4]}) generalized the idea of Gleason's
theorem to observables represented by positive-operator measures.
The resulting Gleason-type theorem (GTT) is much simpler to
prove and it also applies to two-dimensional Hilbert spaces, since
the assumptions being made are stronger than in Gleason's case.
